Metamath Proof Explorer

Theorem olm01

Description: Meet with lattice zero is zero. ( chm0 analog.) (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses olm0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
olm0.m ⊢ ∧ = ( meet ‘ 𝐾 )
olm0.z ⊢ 0 = ( 0. ‘ 𝐾 )
Assertion olm01 ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 0 ) = 0 )

Proof

Step Hyp Ref Expression
1 olm0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 olm0.m ⊢ ∧ = ( meet ‘ 𝐾 )
3 olm0.z ⊢ 0 = ( 0. ‘ 𝐾 )
4 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5 ollat ⊢ ( 𝐾 ∈ OL → 𝐾 ∈ Lat )
6 5 adantr ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ Lat )
7 simpr ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
8 olop ⊢ ( 𝐾 ∈ OL → 𝐾 ∈ OP )
9 8 adantr ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ OP )
10 1 3 op0cl ⊢ ( 𝐾 ∈ OP → 0 ∈ 𝐵 )
11 9 10 syl ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 0 ∈ 𝐵 )
12 1 2 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑋 ∧ 0 ) ∈ 𝐵 )
13 6 7 11 12 syl3anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 0 ) ∈ 𝐵 )
14 1 4 2 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑋 ∧ 0 ) ( le ‘ 𝐾 ) 0 )
15 6 7 11 14 syl3anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 0 ) ( le ‘ 𝐾 ) 0 )
16 1 4 3 op0le ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 𝑋 )
17 8 16 sylan ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 𝑋 )
18 1 4 latref ⊢ ( ( 𝐾 ∈ Lat ∧ 0 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 0 )
19 6 11 18 syl2anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 0 )
20 1 4 2 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) ) → ( ( 0 ( le ‘ 𝐾 ) 𝑋 ∧ 0 ( le ‘ 𝐾 ) 0 ) ↔ 0 ( le ‘ 𝐾 ) ( 𝑋 ∧ 0 ) ) )
21 6 11 7 11 20 syl13anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 ( le ‘ 𝐾 ) 𝑋 ∧ 0 ( le ‘ 𝐾 ) 0 ) ↔ 0 ( le ‘ 𝐾 ) ( 𝑋 ∧ 0 ) ) )
22 17 19 21 mpbi2and ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) ( 𝑋 ∧ 0 ) )
23 1 4 6 13 11 15 22 latasymd ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 0 ) = 0 )